I assume you mean Q(T). t is just the "dummy" variable of integration.

If [tex]Q(t)= \int_0^t (\tauF(\tau)q(p-s\tau))d\tau[/tex]
then the best I can do is
[tex]\frac{dQ}{dt}= tF(t)q(p-st)[/tex]
(by the fundamental theorem of calculus) without knowing what F is. (And is q a constant or a function?)